Local Gibbs sampling beyond local uniformity

Abstract: Local samplers are algorithms that generate random samples based on local queries to high-dimensional distributions, ensuring the samples follow the correct induced distributions while maintaining time complexity that scales locally with the query size. These samplers have broad applications, including deterministic approximate counting [He, Wang, Yin, SODA '23; Feng et al., FOCS '23], sampling from infinite or high-dimensional Gibbs distributions [Anand, Jerrum, SICOMP '22; He, Wang, Yin, FOCS '22], and providing local access to large random objects [Biswas, Rubinfield, Yodpinyanee, ITCS '20]. In this work, we present a local sampler for Gibbs distributions of spin systems whose efficiency does not rely on the "local uniformity" property, which imposes unconditional marginal lower bounds -- a key assumption required by all prior local samplers. For fundamental models such as the Ising model, our algorithm achieves local efficiency in near-critical regimes, providing an exponential improvement over existing methods. Additionally, our approach is applicable to spin systems on graphs with unbounded degrees and supports dynamic sampling within the same near-critical regime.